#495664

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,15,22,16,20,36,19,17,23,35,39,14,21,13,34)(11,24,28,33,26,18,37,30,29,38,40,12,27,31,25,32);(9,11,15,28,16,26,36,37,17,29,35,40,14,27,13,25)(10,18,22,30,20,38,19,12,23,31,39,32,21,24,34,33);(2,5)(6,8)(9,12,14,30,17,33,16,32)(10,11,20,26,23,29,21,27)(13,24,35,31,36,38,15,18)(19,25,39,28,34,37,22,40);(1,3,4,7)(2,6,5,8)(9,13,14,35,17,36,16,15)(10,19,21,22,23,34,20,39)(11,25,27,40,29,37,26,28)(12,24,30,31,33,38,32,18)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=ch,e^b=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402860);